A qualitative approach to the hit problem is that one gives conditions on elements of \((P_t)_n\)  to show that they go to zero in \((QP_t)_n,\)  i.e., belong to \(\sum _{i > 0}{\rm Im}(Sq^{i}).\)  By this, let denote \(\alpha (n)\)  the number of 1's in dyadic expansion of a non-negative integer \(n.\)  It is important to consider the arithmetic function \(\mu : \mathbb {N}\rightarrow \mathbb {N},\)  which is defined by \(\mu (0) = 0\)  and \(\mu (n) = \mbox{min}\lbrace \bullet \in \mathbb {N}:\ \alpha (n + \bullet )\le \bullet \rbrace .\)  Then, Peterson's conjecture [1]}, which has been established by Wood [2]}, points out that \((QP_t)_n = 0\)  when \(\mu (n) > t.\)  We are interested in a relatively efficient tool often used in studying hit problems. It is the \(\mathbb {Z}_2GL_t\) -modules homomorphism \(\widetilde{Sq^0_*}: (QP_t)_{2n+t}\rightarrow (QP_t)_{n},\)  determined by \(\widetilde{Sq^0_*}([f]) = g\)  if \(f = x_1\ldots x_tg^{2},\)  and \(\widetilde{Sq^0_*}([f]) = 0\)  otherwise. This linear map, given by the Kameko thesis [3]}, is related to the above \(\mu \) -function through the following technical result.
